In previous chapters we have usually taken a narrow view of risk as
the variance of return. In this chapter we take a broader view and consider the full distribution of portfolio return. All familiar risk measures,
including variance, value-at-risk (VaR), and expected shortfall, can be
derived from the return distribution. Although the approach to analyzing risk taken in this chapter is much broader, we will see that there are
considerable costs in terms of the statistical reliability of the derived
risk measures.

In section 10.1 we describe the main measures of a return distribution, including the cumulative distribution, the probability density, and
return moments. In section 10.2 we discuss estimation procedures aimed
at measuring the entire return distribution, and in section 10.3 we discuss the estimation procedures designed to measure only the tails of the
return distribution (that is, large losses or gains).

In sections 10.1-10.3 we focus on the univariate case (the return distribution of a single asset or portfolio). In section 10.4 we discuss multivariable dependence structures that are generalizations of covariance,
in order to facilitate multiple-asset analysis of distributions.

10.1 Characterizing Return Distributions

Throughout this section we consider a univariate return r, which we
think of as the random return on some prespecified portfolio such as a
stock or bond index.

10.1.1 The Poor Fit of the Normal Distribution for Return Tails

Recall from chapter 1 that if portfolio return is normally distributed,
then its distribution is completely characterized by mean and variance.
As a risk measure, portfolio variance has the attractive property that it
can be expressed as a quadratic product of the portfolio weight vector

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